Numerical solution of advection-diffusion and convective Cahn-Hilliard equations

Abstract

In this PhD thesis, we construct numerical methods to solve problems described by advectiondiffusion and convective Cahn-Hilliard equations. The advection-diffusion equation models a variety of physical phenomena in fluid dynamics, heat transfer and mass transfer or alternatively describing a stochastically-changing system. The convective Cahn-Hilliard equation is an equation of mathematical physics which describes several physical phenomena such as spinodal decomposition of phase separating systems in the presence of an external field and phase transition in binary liquid mixtures (Golovin et al., 2001; Podolny et al., 2005). In chapter 1, we define some concepts that are required to study some properties of numerical methods. In chapter 2, three numerical methods have been used to solve two problems described by 1D advection-diffusion equation with specified initial and boundary conditions. The methods used are the third order upwind scheme (Dehghan, 2005), fourth order scheme (Dehghan, 2005) and Non-Standard Finite Difference scheme (NSFD) (Mickens, 1994). Two test problems are considered. The first test problem has steep boundary layers near the region x = 1 and this is challenging problem as many schemes are plagued by nonphysical oscillation near steep boundaries. Many methods suffer from computational noise when modelling the second test problem especially when the coefficient of diffusivity is very small for instance 0.01. We compute some errors, namely L2 and L1 errors, dissipation and dispersion errors, total variation and the total mean square error for both problems and compare the computational time when the codes are run on a matlab platform. We then use the optimization technique devised by Appadu (2013) to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by some numerical experiments. In chapter 3, a new finite difference scheme is presented to discretize a 3D advectiondiffusion equation following the work of Dehghan (2005, 2007). We then use this scheme and two existing schemes namely Crank-Nicolson and implicit Chapeau function to solve a 3D advection-diffusion equation with given initial and boundary conditions. We compare the performance of the methods by computing L2- error, L1-error, dispersion error, dissipation error, total mean square error and some performance indices such as mass distribution ratio, mass conservation ratio, total mass and R2 which is a measure of total variation in particle distribution. We also compute the rate of convergence to validate the order of accuracy of the numerical methods. We then use optimization techniques to improve the results from the numerical methods. In chapter 4, we present and analyze four linearized one-level and multilevel (Bousquet et al., 2014) finite volume methods for the 2D convective Cahn-Hilliard equation with specified initial condition and periodic boundary conditions. These methods are constructed in such a way that some properties of the continuous model are preserved. The nonlinear terms are approximated by a linear expression based on Mickens' rule (Mickens, 1994) of nonlocal approximations of nonlinear terms. We prove the existence and uniqueness, convergence and stability of the solution for the numerical schemes formulated. Numerical experiments for a test problem have been carried out to test the new numerical methods. We compute L2-error, rate of convergence and computational (CPU) time for some temporal and spatial step sizes at a given time. For the 1D convective Cahn-Hilliard equation, we present numerical simulations and compute convergence rates as the analysis is the same with the analysis of the 2D convective Cahn-Hilliard equation.